OPEN PROBLEMS IN TOPOLOGY

§3] Non-metrizable Normal Moore Spaces 43On the other hand, a theorem would be quite surprising and would provideconvincing evidence that the Moore plane is canonical.3. Non-metrizable Normal Moore SpacesProblem 11. (Palermo #8; Tall’s A1) Does 2 ℵ0 = ℵ 2 imply the existence of 79. ?a non-metrizable normal Moore space?I only add that it is dangerous to spend 95% of the effort on a questiontrying to prove it in one direction. Very little effort has gone into trying tomodify Fleissner’s [1982b, 1982a] construction of a non-metrizable Moorespace from the continuum hypothesis. In fact, there’s probably only abouttwo or three people who really understand his construction (I am not one ofthem). If the conventional wisdom that collapsing a large cardinal should getthe consistency of the normal Moore space conjecture with 2 ℵ0 = ℵ 2 ,thenwhy hasn’t it been done?Problem 12. (Palermo #9; Tall’s A3) Does the existence of a non-metrizable 80. ?normal Moore space imply the existence of a metacompact non-metrizablenormal Moore space?Problem 13. (Palermo #12) Does the existence of a non-metrizable nor- 81. ?mal Moore space imply the existence of a normal Moore space which is notcollectionwise normal with respect to metrizable sets?The reason this question remains of interest is that a tremendous amountof effort has gone into obtaining partial positive results. Rudin and Starbird[1977] and Nyikos [1981] both obtained some technical results of greatinterest. Nyikos showed, in particular, that if there is a non-metrizable normalMoore space, then there is a metacompact Moore space with a family of closedsets which is normalized but not separated. In Watson [19∞a], it is shownthat if there is a non-metrizable normal Moore space which is non-metrizablebecause it has a nonseparated discrete family of closed metrizable sets thenthere is a metacompact non-metrizable normal Moore space. This means thatif you believe in a counterexample you had better solve problem 32 first! Ifyou believe in a theorem as Rudin and Starbird and Nyikos did, you have alot of reading to do. I think there is a counterexample.Problem 14. (Palermo #10; see Tall’s A2 and A4) Does the existence of 82. ?a non-metrizable normal Moore space imply the existence of a para-Lindelöfnon-metrizable normal Moore space?The normal Moore space problem enjoyed lots of consistent counterexampleslong before para-Lindelöf raised it’s head. However in [1981] Caryn